Root systems and root lattices in number fields

The following construction of the root system of type $G_2$ is known (see the book by J.-P. Serre "Lie algebras and Lie groups", p. 315, Russian translation): “It can be described as a set algebraic integers of the cyclotomic field generated by the cube root of unity, with the norm 1 or 3". This talk, based on the joint results with Yu. G. Zarhin, is devoted to realization of root systems, their Weyl groups and their root lattices in the form of subsets, groups and lattices, naturally associated with fields of algebraic numbers.